Correlation%20and%20Regression

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Correlation and Regression
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Relationships between variables

• Example Suppose that you notice that the more
  you study for an exam, the better your score
  typically is.
• This suggests that there is a relationship
  between study time and test performance.
• We call this relationship a correlation.

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Relationships between variables

• Properties of a correlation
• Form (linear or non-linear)
• Direction (positive or negative)
• Strength (none, weak, strong, perfect)
• To examine this relationship you should
• Make a scatterplot
• Compute the Correlation Coefficient

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Scatterplot

• Plots one variable against the other
• Useful for seeing the relationship
• Form, Direction, and Strength
• Each point corresponds to a different individual
• Imagine a line through the data points

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Scatterplot
Hours study X Exam perf. Y
6 6
1 2
5 6
3 4
3 2
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Correlation Coefficient

• A numerical description of the relationship
  between two variables
• For relationship between two continuous variables
  we use Pearsons r
• It basically tells us how much our two variables
  vary together
• As X goes up, what does Y typically do
• X?, Y?
• X?, Y?
• X?, Y?

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Form
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Direction
Negative
Positive

• As X goes up, Y goes up
• X Y vary in the same direction
• Positive Pearsons r

• As X goes up, Y goes down
• X Y vary in opposite directions
• Negative Pearsons r

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Strength

• Zero means no relationship.
• The farther the r is from zero, the stronger the
  relationship
• The strength of the relationship
• Spread around the line (note the axis scales)

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Strength
r -1.0 perfect negative corr.
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Strength
Rel A
Rel B
Which relationship is stronger? Rel A, -0.8 is
stronger than 0.5
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Regression

• Compute the equation for the line that best fits
  the data points

Y (X)(slope) (intercept)
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Regression

• Can make specific predictions about Y based on X

X 5 Y ?
Y (X)(.5) (2.0)
Y (5)(.5) (2.0) Y 2.5 2 4.5
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Regression

• Also need a measure of error

Y X(.5) (2.0) error
Y X(.5) (2.0) error

• Same line, but different relationships (strength
  difference)

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Cautions with correlation regression

• Dont make causal claims
• Dont extrapolate
• Extreme scores (outliers) can strongly influence
  the calculated relationship